%\gray
\section{Sheaves for a Grothendieck Topology}
%!%$http://en.wikipedia.org/wiki/Grothendieck_topology$\\
%!%$http://ncatlab.org/nlab/show/Grothendieck+pretopology$
\subsection{Grothendieck Topology}
\begin{definition}[Grothendieck Pre-topology]\cite{Lev07}
\noindent The concept of sites, a category endowed with Grothendieck toplogy, is a generalisation of the concept of topological spaces in such a way that enables the development of the theory of sheaves on sites as an analogue of the theory of sheaves on topological spaces. Hence, it enables the the development of cohomology theory on these sites. Based on Serre's observation of a covering with non-monomorphism of algebraic groups, and Grothendieck observation of the analogy between the fundamental group of covering spaces, and Galois group of field extensions, Grothendiek generalise the concept of covering to be a family of 'general' morphisms that satisfies some conditions that makes the development of sheaf theory possible.Note that, the definition of a sheaf does not require any additional conditions on the family of morphisms in the covering.

\noindent \tcb{Explain the importance of refinement} Grothendiek topology is defined so that 'potential covering' (sieves) that are a refinement of a Grothendiek covering is also a Grothendiek covering.


Let $\bcC$ be a category, then we say that the collection $\tau=\{Cov_{\tau}(U)\subseteq 2^{Ob((\bcC \downarrow U))}:U\in \bcC\}$ is a Grothendieck pre-topology on $\bcC$, we call elements of $Cov_{\tau}(U)$ coverings of $U$ with respect to $\tau$, and  iff it satisfies:
\begin{itemize}
\item Identity axiom: $\forall U\in \bcC, \{id_U\}\in Cov_{\tau}(U)$.
\item Stability axiom: $\forall f:V\rightarrow U,\bcU=\{f_i:U_i\rightarrow U:i \in I\}\in Cov_{\tau}(U)$ , then the family $f^{\ast}\bcU\:=\{p_{2,U_i,V}:U_i\times_{U}V\rightarrow V|i \in I\}$ exists and is a covering of $V$, where $p_{2,U_i,V}$ is the pullback projection on $V$, for each $i \in I$. \tcb{We may refer to this axiom as stability under base change.}
\item Transitivity axiom: $\forall \bcU=\{f_i:U_i\rightarrow U|i \in I\}\in Cov_{\tau}(U), 
\bcU_i=\{g_{ij}:U_{ij}\rightarrow U_i|j\in J_i\}\in Cov_{\tau}(U_i)$ for $i \in I$, then $\bcU\{\bcU_i|i\in I\}:=\{f_i\circ g_{ij}|j\in J_i, i\in I\}\in Cov_{\tau}(U)$. \tcb{We may refer to this axiom as stability under composition.}
\end{itemize}
\end{definition}
\begin{remark}
We notice that grothendieck pre-topology is a generalization of topological open covering. That, for a topolological space $(X,\tau_X)$ and $U\in \tau_X$, we notice that $U$ is an open covering of itself, open subsets inherit the coverings of there containing open sets, by the mean of intersection (the pullback in $\tau_X$), and a collection of coverings of elements of covering of $U$ forms a covering of $U$. Hence, the collection of topological open coverings equips the preordered category $\bcO(\tau_X)$ with Grothendieck pre-topology, denoted by $\tau_X$, in order to distinguish it from other possible pretopologies on $\bcO(\tau_X)$. However, Grothendieck covering on $\bcO(\tau_U)$ is not necessary a topological covering. That, for example in the Grothendieck pre-topology \{$2^{Ob((\bcO(\tau_X) \downarrow U))}:U\in \tau_X\}$, the set $\{\emptyset\stackrel{\emptyset}{\rightarrow} U\}$ forms a covering for $U$ in the sense of Grothendieck pre-topology, although it is not a topological open covering.
\end{remark}
\begin{terminology}
From now, we will abuse notion and refer to a Grothendieck pre-topology on a category as a pre-topology on a category.
\end{terminology}
\begin{lemma}
Let $\bcC$ be a category, then:
\begin{itemize}
\item Let $\tau_0(\bcC)=\{\{id_U\}:U\in \bcC\}$. For any $f:V\rightarrow U$ in $\bcC$, the diagram $\xymatrix{&V\ar[d]^{f}\\
U\ar[r]_{id_U}&U}$ has the pullback diagram 
$\xymatrix{V\ar[r]^{id_V}\ar[d]_{f}&V\ar[d]^{f}\\
U\ar[r]_{id_U}&U}$. Hence, one can readily see that $\tau_0(\bcC)$ forms a pre-topology on $\bcC$, called the intial pre-topology on $\bcC$ \tcb{justify the name!}. $\tau_0(\bcC)$ might be referred to by the indiscrete topology on $\bcC$ and denoted $ind(\bcC)$.
\item Let $\bcC$ has all pullbacks, then $\tau_1(\bcC)=\{2^{ob((\bcC\downarrow U))}:U\in \bcC\}$ forms a pre-topology on $\bcC$, called the final pre-topology on $\bcC$ \tcb{justify the name!}.
\end{itemize}
\end{lemma}
\begin{counterexample}
$\tau_1(\bcC)$ does not necessary exist for arbitrary category $\bcC$. For instance, let $\bcC$ be the category generated by the diagram:
%??!!
$$\xymatrix{W\ar[r]^{g}\ar[rdd]_{h}&V\ar[rd]^{f}&\\
&&U\\
W'\ar[r]_{h'}\ar[ruu]_{g'}&V'\ar[ru]_{f'}&}$$
with the relations $f\circ g=f'\circ g'$ and $f\circ h=f'\circ h'$. Then, we easily notice that the diagram $V \stackrel{f}{\rightarrow} U \stackrel{f'}{\leftarrow} V'$ does not have a pullback. Hence, $\tau_1(\bcC)$ is not a pre-topology on $\bcC$.
\end{counterexample}
\begin{lemma}
Let $\tau$ be a pre-topology on $\bcC$, and $f:V\rightarrow U$ is an isomorphism in $\bcC$ then $\{f\}\in Cov_{\tau}(U)$.
\end{lemma}
\begin{proof}
Based on the identity axiom $\{id_U\}\in Cov_{\tau}(U)$, $f:V\rightarrow U$ in $\bcC$, then the stability axiom implies $\{f=(id_U\times_U f)_V \}\in Cov_{\tau}(U).$
\end{proof}
Since all $id_U, U\in \bcC$ are isomorphisms, then notation of Grothendieck pre-topology could be given by replacing the identity axiom by requiring that $\{f\}\in Cov_{\tau}(U)$, for all isomorphisms $f:V\rightarrow U$ in $\bcC$, and calling it the isomorphism.
\begin{definition}
Let $\tau,\tau'$ be pre-topologies on the the categories $\bcC,\bcC$, respectively. We define the morphism of Grothendick pre-topologies $\bcF:\tau\rightarrow\tau'$ to be a functor $\bcF:\bcC\rightarrow \bcC'$ that satisfies:
\begin{itemize}
\item $\forall U\in \bcC,\forall \{f_i:U_i\rightarrow U,i\in I\} \in Cov_U$ implies $\{\bcF(f_i):\bcF(U_i)\longrightarrow \bcF(U),i\in I\} \in Cov_{\bcF(U)}$. I.e. $\bcF$ maps coverings to coverings. \tcb{What is the significance of this property? Where is it being used?}.
\item $\forall U\in \bcC,\forall \{f_i:U_i \rightarrow U,i\in I\} \in Cov_U, f:V\rightarrow U$ in $\bcC$ the canonical morphism $\bcF(U_i\times_U V)\rightarrow \bcF(U_i)\times_{\bcF(U)}\bcF(V)$ is an isomorphism in $\bcC',\forall i\in I$.
$$\xymatrix{\bcF(U_i\times_U V)\ar@{-->}[rd]\ar@/^/[drrr]^{\bcF(f)}\ar@/_/[dddr]_{\bcF(f_i)}&&&\\
&\bcF(U_i)\times_{\bcF(U)} \bcF(V)\ar[rr]^{\pi_{\bcF(V)}}\ar[dd]_{\pi_{\bcF(U_i)}}&&\bcF(V)\ar[dd]^{\bcF(f)}\\\\
&\bcF(U_i)\ar[rr]_{\bcF(f_i)}&&\bcF(U)}$$
\tcb{Why? is it to identify the change of bases of both topologies, of course within the image of $\bcF$? On the topological space it works as gluing condition.}
\end{itemize}
\end{definition}
\begin{example}[Examples of Grothendieck pre-topologies and morphisms]\tcr{How to move to next line here with out extra typing??}
\begin{itemize}
\item Let $(X,\bcO(\tau_X)),(X',\bcO(\tau_{X'}))$ be topological spaces. Then, one can readily show that $\bcF:\tau\rightarrow \tau'$ is a morphism of Grothendieck pre-topologies iff $\bcF=f^{-1}$ for some continuous map$f:X'\rightarrow X$.
\item \tcb{Get back to the three examples of canonical pre-topology on a category with all pullbacks}.
\end{itemize}
\end{example}
\tcb{Construct the category of pre-topologies on a category with pullbacks!}.
\begin{terminology}
From now on, $\bcC_{\tau}$ will denote a category $\bcC$ equipped with Grothendieck pre-topology $\tau$.
\end{terminology}
\begin{definition}[Pre-sheaves]
Let $\bcC, \bcA$ be categories, an $\bcA$-valued pre-sheaf on $\bcC$ is a functor $\bcP:\bcC^{op}\rightarrow \bcA$.
\end{definition}
\begin{example}[Natural Example of pre-sheaf]
Let $\bcC$ be a (small) category, $X\in \bcC$, then $Hom_{\bcC}(-,X):\bcC^{op}\rightarrow \Sets$ is a pre-sheaf of sets (sets-valued pre-sheaf). Pre-sheaves defined this way are called representable pre-sheaves.
\end{example}
\begin{definition}[Sieves]
Let $\bcC$ be a (small) category, $X\in \bcC$ we define a sieve $S$ on $X$ to be a subfunctor $S\hookrightarrow Hom_{\bcC}(-,X):\bcC^{op}\rightarrow \Sets$.
\end{definition}
Then, every sieve of $X\in \bcC$ is a pre-sheaf of sets on $\bcC$.\\
\tcb{what is the advantage of having the subfunctor? i.e. the monomorphism? rather than being able to identify elements of $S(U)$ with elements of $hom_{\bcC}(U,X)$?}
\begin{remark}
Let $S$ be a sieve on $X\in \bcC$, then $\forall U\in \bcC, S(U) \stackrel{i_U}{\hookrightarrow} Hom_{\bcC}(U,X)$, and $\forall f:V\rightarrow U$ in $\bcC$, the following diagram commutes:\\
$\xymatrix{
V\ar[d]^f&S(V)\ar@{^(->}[r]^{i_V}&Hom_{\bcC}(V,X)\\
U&S(U)\ar@{^(->}[r]_{i_U}\ar[u]^{S(f)}&Hom_{\bcC}(U,X)\ar[u]_{(f^{\ast},(id_U)_{\ast})}}$\\
Hence, for any morphism $g:U\rightarrow X, g\in S(U)$ , the morphism $g\circ f\in S	(V)$, which is usually called saturation condition. \\
\tcb{read the last conditions}
\end{remark}
\begin{definition}
Let $S,S'$ be sieves on $X,X'\in \bcC$, respectively. Then, we define the sieves morphism $\varphi:S\rightarrow S'$ to be a morphism of functors (natural transformations). Then, we denote the category of sieves on objects of $\bcC$ and their morphisms by $\Svs(\bcC)$, and, we define $i:\Svs(\bcC)\rightarrow \Sets$ such that $i(S):=\displaystyle \bigsqcup_{U\in \bcC}S(U)$, and for $\varphi:S\rightarrow S', i(\varphi):i(S)\rightarrow i(S')$ such that $i(\varphi)(s)=\varphi_U(s),$ for $s\in S(U)$.
\end{definition}
\begin{lemma}
Let $\bcC$ be category then $i:\Svs(\bcC)\rightarrow \Sets$, defined above, then $i$ is a monic functor.
\end{lemma}
\begin{proof}
\tcb{type the proof!}
\end{proof}
\begin{question}
\tcb{does $i$ have an adjunction? Can the generated sieve be defined as an image of such adjunction?}.
\end{question}
\begin{remark}
Since $i$ is an injection on objects then we can identify $S$ with $S(\bcC):=i(S)$.
\end{remark}
\begin{question}
For any sieve $S_X$ on $X\in \bcC_{\tau}$, then a natural question arise! What are the conditions $S_X$'s have to satisfy to form singleton Grothendieck pre-topology $\tau:=\{\{S_X(\bcC)\}:X\in \bcC\}$ on $\bcC$?
\end{question}
\begin{definition}[\tcb{Another approach}]
Let $S,S'$ be sieves on $X\in \bcC$. Then, we define the sieves morphism $\varphi:S\rightarrow S'$ to be a morphism of functors (natural transformations). Then, we denote the category of sieves on $X\in \bcC$ and their morphisms by $\Svs(X)$, and, we define $i:\Svs(X)\rightarrow \mathfrak{2}^{(\bcC\downarrow X)}$ such that $i(S):=\displaystyle \bigsqcup_{U\in \bcC}S(U)$, and for $\varphi:S\rightarrow S', i(\varphi):i(S)\rightarrow i(S')$ such that $i(\varphi)(s)=\varphi_U(s),$ for $s\in S(U)$.
\end{definition}
\tcb{Try to define the generated sieve to be the minimum (w.r.t inclusion) image of a sieve that contain the set!}
\begin{lemma}
Let $\{f_i:X_i\rightarrow X, i\in I\}$ be a collection of morphisms in a category $\bcC$, and let $S_{\bcC}:=\{f:U\rightarrow X | \exists i\in I, \exists g:U\rightarrow X_i:f=f_i\circ g \}$, $S:\bcC\rightarrow \Sets$, defined by $S(U)=\{f:f\in S_{\bcC}, \dom(f)=U \}$, and $S(h)=(h^{\ast},(id_X)_{\ast})|_{S(U)}$ for $h:V\rightarrow U$ in $\bcC$. Then, $S$ is a sieve on $U$.
\end{lemma} 
\begin{proof}
One can readily see that $S$ is a functor, and that since $\forall U\in \bcC:S(U)\subseteq Hom_{\bcC}(U,X)$ and $S(h)=(h^{\ast},(id_X)_{\ast})|_{S(U)}$ for $h:V\rightarrow U$, then $S$ is a subfunctor of $Hom_{\bcC}(-,X)$ given by the inclusion natural transformation.
\end{proof}
\begin{definition}[Generated Sieves]
Let $\{f_i:X_i\rightarrow X, i\in I\}$ be a collection of morphisms in a category $\bcC$. Then, we call $S:\bcC\rightarrow \Sets$, defined in the previous lemma, the sieve, on $X$, generated by $\{f_i:X_i\rightarrow X, i\in I\}$.
\end{definition}
\begin{definition}[Pullback of Sieves]\label{Pullback of Sieves}
Let $S$ be a sieve on $X\in \bcC$, and $f:Y\rightarrow X$ in $\bcC$, then we define the restriction of $S$ to $Y$ to be the sieve generated by $\{g_i:Y_i\rightarrow Y| f\circ g_i\in S(Y_i)\}$,denoted it by $\S_Y$.
\end{definition} 
\begin{remark}
In the settings of definition \ref{Pullback of Sieves} , for every commutative diagram:
$$
\xymatrix{V\ar[r]^{g'}\ar[d]_h&Y\ar[d]^f\\
U\ar[r]_g&X}
$$
where $g\in S(U)$ , then $g'\in S_Y(V)$.\\
We also notice that if $f:Y\rightarrow X\in S(Y)$, then $S_Y=Hom_{\bcC}(-,Y)$.
\end{remark}
\begin{remark}
We notice that, for the sieve $S$ on $X\in \bcC$, $S(\bcC)$ does not necessary form a covering for $X$, in the sense of pre-topology, that $S(X)$ is allowed to be $\emptyset$, also $id_X$ does not have to belong to $S(X)$. \tcb{Other conditions cannot be checked for one covering!}.\\
\tcb{Below, we give the definition of Grothendieck topology, which is the conditions a collection of sieves have to adhere to to in order to form a Grothendieck pre-topology.}
\end{remark}
\begin{definition}[Grothendieck Topology]
Let $\bcC$ be a category, then we say that the collection $\tau=\{S_{\tau}(U)| S_{\tau}(U) \subset ob(\Svs(U)), U\in \bcC\}$ is a Grothendieck topology and we call elements of $S_{\tau}(U)$ covering sieves of $U$ with respect to $\tau$ iff it satisfies:
\begin{itemize}
\item (The maximal sieve) $\forall U\in \bcC$, the sieve generated by $\{id_U\}$ is in $S_{\tau}(U)$, i.e. $Hom_{\bcC}(-,U)\in S_{\tau}(U)$.\\
\tcb{Here we basicly want to consider objects of $\bcC$ with morphisms to U to be open subsets of the space $U$. Still, it does not guarantee the first condition of pre-topology}.
\item (Stability axiom) $\forall S\in S_{\tau}(U), V \in \bcC$ such that $\exists f:V\rightarrow U$, then $S_V\in S_{\tau}(V)$.\\ I.e. the collection of covering sieves is closed under pullback of sieves (restriction) \tcb{show it as a categorical pullback! $S\times_{Hom_{\bcC}(-,U)}Hom_{\bcC}(-,V)$} \\
\tcb{It does not guarantee the existence of the pullback, it's the equivalence of inheriting the topology (pullback of topology)}.
\item (Transitivity axiom) Let $S\in S_{\tau}(U), T$ a sieve on $U$ such that for every $f:V\rightarrow U\in S(V), T_V\in S_{\tau}(V)$, then $T\in S_{\tau}(U)$.\\
\tcb{Is it gluing condition? or uniqueness condition? It's somehow equivalent to the third condition of pre-topology.}\\
\tcb{Read this condition}.
\end{itemize}
\end{definition}
\noindent \tcb{why is it taken along  a covering seive? not any morphism or all?}
\begin{definition}[Site]
Let $\bcC$ be category with a Grothendieck topology $\tau$, then we call $(\bcC,\tau)$ a site, and we denote it by $\bcC_{\tau}$.
\end{definition}
\begin{counterexample}[Grothendieck pre-topology does not necessary form Grothendieck topology]
\tcb{...Type an example}
\end{counterexample}
\begin{counterexample}[Grothendieck topology does not necessary form Grothendieck pre-topology]
\tcb{...Type an example}
\end{counterexample}
\begin{remark}[Generation of Grothendieck topology]
Given Grothendieck pre-topology $\tau$ on $\bcC$, then for all $\bcU=\{f_i:U\rightarrow U| i\in I\} \in Cov_{\tau}(U)$ we consider the sieve $S_{\bcU}$ generated by $\bcU$. Then we obtain a family of sieves $\tilde{S}(U)$ for every $U\in \bcC$. \tcb{Then, show that the collection $\{\tilde{S}(U)|U\in \bcC\}$ generate Grothendieck topology on $\bcC$.}
\end{remark}
%!% Sieves are fully faithfull and discrete fibration.
\tcb{what if the category has loops?}
\tcb{the topology is not a pretopology}
\tcb{Why is it called sieve?}
\tcb{What is the advantage of dealing with effictive epimorphisms? What is effective about them?}\\


\tcb{mention why you are defining the category of covering of an object!}
\begin{definition}[Refinement Morphisms]
Let $\bcC_{\tau}$ be a site, and $\bcU=\{f_i:U_i\rightarrow U|i\in I\}, \bcV=\{g_j:V_j\rightarrow U|j\in J\}\in Cov_{\tau}(U), U\in \bcC$. We define a refinement morphism $\phi_{\epsilon}:\bcV\rightarrow \bcU$ to be a pair $\phi_{\epsilon}=(\epsilon,\{\phi_j)|j\in J\}$), where $\epsilon:J\rightarrow I$ is a map of sets, and $\forall j\in J$, $\phi_j:V_j\rightarrow U_{\epsilon(j)}$ is a $U$-mporphism in $\bcC$, i.e. it makes the following diagram commutate:
$$\xymatrix{V_j\ar[rr]^{\phi_j}\ar[rd]_{g_j}&&U_{\epsilon(j)}\ar[ld]^{\epsilon(j)}\\
&U&}$$
Then, we say that $\bcV$ is a refinement of $\bcU$
\end{definition}
\begin{definition}
Let $\bcC_{\tau}$ be a site, $\xymatrix{\bcW\ar[r]^{\Psi_{\delta}}&\bcW\ar[r]^{\phi_{\epsilon}}&\bcU}$ be refinement morphisms for $\bcW,\bcV,\bcU \in Cov_{\tau}(U):U\in \bcC$. Then we define the composition $\phi_{\epsilon}\circ \Psi_{\delta}:=(\epsilon\circ\delta,\{\phi_{\delta(k)}\circ\psi_k|k\in K\})$, where $K$ is the set of indecies of $\bcW$.
\end{definition}
\begin{lemma}
Composition of refinement morphisms is a refinement morphism, it is associative, and every covering covering admits the evident identity refinement morphism with respect to this composition.
\end{lemma}
\begin{proof}
Trivial
\end{proof}
\begin{definition}
Let $\bcC_{\tau}$ be a site, $U\in \bcC$, we define $C_U$ to be the category of $Cov_{\tau}(U)$ and the refinement morphisms of coverings of $U$. We call $C_U$ the category of covering of $U$.
\end{definition}
\begin{lemma}\label{CommonRef}
Let $\bcC_{\tau}$ be a site, $U\in \bcC$, the category $C_U$ is a filtering category. Moreover, $C_U$ has all binary products.
\end{lemma}
\begin{proof}
Let $\bcU=\{f_i:U_i\rightarrow U|i\in I\},\bcU'=\{f'_{i'}:U'_{i'}\rightarrow U|i'\in I'\}\in C_U$, then:\\
$\forall i\in I, f_i^{\ast}\bcU'\in C_{U_i}$, hence $\bcU\{f_i^{\ast}\bcU'|i\in I\}\in C_U$\\
Also, $\forall i'\in I', f_{i'}^{'\ast}\bcU\in C_{U_{i'}}$, hence $\bcU'\{f_{i'}^{'\ast}\bcU|i'\in I'\} \in C_U$.\\
Notice that $\bcU\{f_i^{\ast}\bcU'|i\in I\}=\bcU'\{f_{i'}^{'\ast}\bcU|i'\in I'\}$. We define, $Com(\bcU,\bcU'):=\bcU\{f_i^{\ast}\bcU'|i\in I\}$. Then, $Com(\bcU,\bcU')$ is a common refinement of $\bcU$ and $\bcU'$ with the refinement morphisms:
$$
{\small \xymatrix{&Com(\bcU,\bcU')\ar[dl]_{\Pr_1}\ar[dr]^{\Pr_2}&\\
\bcU&&\bcU'}}
$$
Where $\Pr_l:=(P_{l,I,I'},\{p_{l,U_i,U'_{i'}}|(i,i')\in I\times I'\})$, such that $P_{1,I,I'}:I\times I'\rightarrow I$
and $P_{2,I,I'}:I\times I'\rightarrow I'$ are the product projections. Based on the definition of $Com(\bcU,\bcU')$, one can readily see that $\Pr_l$'s are refinement morphisms.\\
Let $\bcV=\{g_j:V_j\rightarrow U|j\in J\}$ be a common refinement for $\bcU$ and $\bcU'$, i.e. having the following diagram in $C_U$:
$${\small
\xymatrix{&\bcV\ar[dl]_{\psi_{\delta}}\ar[dr]^{\psi'_{\delta'}}&\\
\bcU&&\bcU'}}
$$
Let $\sigma:J\rightarrow I\times I'$ be defined by $\sigma=(\delta,\delta')$. Then, based on the universal property of $U_{\delta(j)}\times_U U'_{\delta'(j)}$, $\exists!\ \xi_j:V_j\rightarrow U_{\delta(j)}\times_U U'_{\delta'(j)}$ that makes the following diagram commute, $\forall j\in J$:
$$
\xymatrix{
&&&&U_{\delta(j)}\ar[rrd]|{f_{\delta(j)}}\\
V_j\ar@{-->}[rr]|{\xi_j}\ar@/^/[rrrru]|{\psi_j}\ar@/_/[rrrrd]|{\psi'_j}&&U_{\delta(j)}\times_U U'_{\delta'(j)}\ar[rru]|{p_{1,\delta(j),\delta'(j)}}\ar[rrd]|{p_{2,\delta(j),\delta'(j)}}&&&&U\\
&&&&U_{\delta(j)}\ar[rru]|{f'_{\delta'(j)}}
}
$$
That the external diagram is commutative because $f_{\delta(j)}\circ \psi_{j}=g_g=f'_{\delta'(j)}\circ \psi'_{j}$.\\
Then, straightforward calculations shows that $\xi_{\sigma}=(\sigma,\{\xi_{i''}|i''\in I''\})$ is the unique refinement morphism that makes the following diagram commutative.
$$
\xymatrix{
&&&&\bcU\\
\bcU''\ar@{-->}[rr]|{\xi_{\sigma}}\ar@/^/[rrrru]|{\psi_{\delta}}\ar\ar@/_/[rrrrd]|{\psi'_{\delta'}}&&Com(\bcU,\bcU')\ar[rru]|{Pr_1}\ar[rrd]|{Pr_2}\\
&&&&\bcU'
}
$$
I.e. $Com(\bcU,\bcU')$ is the product $\bcU\times \bcU'$ in $C_U$.
\end{proof}

\begin{lemma}\label{RefTh}
Let $\bcV=\{g_j:V_j\rightarrow U|j\in J\}\in \Cov(U)$, and $\bcV'_j=\{g'_{jj'}:V'_{jj'}\rightarrow V_j|j'\in J'_j\}\in \Cov(V_j),\forall j \in J$, let $\bcV':=\bcV\{\bcV_j|j\in J\}=\{h'_{jj'}:=g_j\circ g'_{jj'}|j\in J,j'\in J'_j\}$. Then:
\begin{itemize}
\item There is a refinement morphism $\psi_{\epsilon}:\bcV'\rightarrow\bcV$. Hence, $\forall j\in J$, there is a refinement morphism $g_j^{\ast}\psi_{\epsilon}:g_j^{\ast}\bcV'\rightarrow g_j^{\ast}\bcV$ in $\Cov(V_j)$
\item $\bcV$ is a refinement of $\{id_U\}$.
\item $\forall j\in J$, there is a refinement morphism $\Delta_{j,inc_{j}}:\{id_{\bcV_j}\}\rightarrow g_j^{\ast}\bcV$.
\item $\forall j\in J$, there is a refinement morphism $\phi_{j,id}:\bcV'_j\rightarrow g_j^{\ast}\bcV'$ in $\Cov(V_j)$. \tcb{could be proven as a result of the last two points.}
\item If $g_j$ is a monomorphism, $\bcV'_j $ is isomorphic to $ g_j^{\ast}\bcV'$ in $\Cov(V_j)$. \tcb{check if it's iff condition}
\item $\forall j\in J, j'\in J'_j$ there is a refinement morphism $\xi_{jj',id}:g_{jj'}^{'\ast}\bcV'_j\rightarrow h_{jj'}^{'\ast}\bcV'$ in $\Cov(V'_{jj'})$. \tcb{What about isomorphism?}
\item $\forall j\in J$, the following diagram commutes:
$$
\xymatrix{\bcP^+(U)\ar[rrr]^{\bcP^+(g_j)}&&&\bcP^+(V_j)\\
\bcP_U(\bcV')\ar[u]^{i_{\bcV'}}\ar[rrr]_{\bcP_{V_j}(\phi_{j,id}) \circ \bcP_{g_j}(\bcV')}&&&\bcP_{V_j}(\bcV'_j)\ar[u]_{i_{\bcV'_j}}
}
$$
\end{itemize}
\end{lemma}
\begin{proof}Let $J':=\displaystyle\bigsqcup_{j\in J}\{j\}\times J'_j$, $\epsilon:J'\rightarrow J$ be the projection on $J$, given by $\epsilon(j,j')=j$:
\begin{itemize}
\item $\forall (j,j')\in J'$ let $\psi_{jj'}:=g'_{jj'}, \forall (j,j')\in J'$, which is $U$-morphisms, based on the definition of $g'_{jj'}$. Hence, $\psi_{\epsilon}=(\epsilon,\{\psi_{jj'}|(j,j')\in J'\}):\bcV'\rightarrow \bcV$ is a refinement morphism.\\
\ref{fastphiL} implies that, $\forall j \in J$, $g_j^{\ast}\psi_{\epsilon}:g_j^{\ast}\bcV'\rightarrow g_j^{\ast}\bcV$ is a refinement morphism in $\Cov(V_j)$.
%Where $\psi_{jj'}:V'_{jj'}\times_U V_j\rightarrow V_j\times_U V_j$ that makes the following diagram commute, $\forall j' \in J'_j$:
%$$ \xymatrix{&&V'_{jj'}\ar[r]^{g'_{jj'}}&V_j\ar@/^/[dr]^{g_j}\\V'_{jj'}\times_U V_j\ar@/^/[urr]^{p_1}\ar@/_/[drr]_{p_2}\ar@{-->}[rr]|{\psi_{jj'}}&&V_j\times_U V_j\ar[ur]|{p_1}\ar[dr]|{p_2}&&U\\&&V_j\ar@{=}[r]&V_j\ar@/_/[ur]_{g_j}}$$
%The commutativity of the lower partial diagram implies $\psi_{j,\epsilon}=(\epsilon,\{\psi_{jj'}|j'\in J'_j\}):g_j^{\ast}\bcV'\rightarrow g_j^{\ast}\bcV$ is a refinement morphism.
\item $\forall j \in J, g_j$ is a $U$-morphism, hence the $\bcV_{\{\emptyset\}}:\bcV\rightarrow \{id_U\}$ is a refinement morphism, where .
\item $\forall j \in J$, let $\Delta_j:V_j\rightarrow V_j\times_U V_j$ be the unique such morphism that makes the following diagram commutative:
$$
\xymatrix{
&&&V_j\ar@/^/[dr]^{g_j}\\
V_j\ar@/^/[urrr]^{id_{V_j}}\ar@/_/[drrr]_{id_{V_j}}\ar@{-->}[rr]|{\Delta_j}&&V_j\times_U V_j\ar[ur]|{p_1}\ar[dr]|{p_2}&&U\\
&&&V_j\ar@/_/[ur]_{g_j}
}
$$
The commutativity of lower left partial diagram makes $\Delta_{j,inc_j}=(inc_j,\{\Delta_j\}):\{id_{\bcV_j}\}\rightarrow g_j^{\ast}\bcV'$ a refinement morphism. Where $inc_j:\{j\}\rightarrow J\times \{j\}$ is the inclusion map, given by $inc_j(j)=(j,j)\in J\times \{j\}$.
\item $\forall j \in J$, we have the refinement morphisms $\phi_{j,id}:\bcV'_j\rightarrow g_j^{\ast}\bcV'$, where $\phi_{jj'}:V'_{jj'}\rightarrow V'_{jj'}\times_U V_j$ is the unique such morphism that makes the following diagram commutative, $\forall j'\in J'_j$:

$$
\xymatrix{
&&&V'_{jj'}\ar@/^/[rd]^{g'_{jj'}}\\
V'_{jj'}\ar@{-->}[rr]|{\phi_{jj'}}\ar@/^/[rrru]^{id_{V'_{jj'}}}\ar@/_/[rrrd]_{g'_{jj'}}&&V'_{jj'}\times_U V_j\ar[ur]|{p_1}\ar[dr]|{p_2}&&U\\
&&&V_j\ar@/_/[ru]_{g_j}
}
$$
The commutative of the external diagram is guaranteed by the definition of $g'_{jj'}$. And the commutativity of the lower left-hand partial diagram implies that $\phi_{j,id}$ is a refinement morphism.
\item Consider the following diagram, $\forall j'\in J'_j$:
$$
\xymatrix{
V'_{jj'}\times_U V_j\ar[r]^{p_2}\ar[d]_{p_1}&V_j\ar[d]^{g_j}\\
V'_{jj'}\ar[r]_{g'_{jj'}}\ar[ur]|{g'_{jj'}}&U
}
$$

We notice that the external diagram, and the lower right triangle are commutative, hence, $g_j\circ (g'_{jj'}\circ p_1)=g_j\circ p_2$. If $g_j$ is a monomorphism, then $g'_{jj'}\circ p_1=p_2$, i.e. the above diagram commute, and $p_1$ is a $V_j$-morphism. Hence, we have the refinement morphism $\phi'_{j,id}:g_j^{\ast}\bcV'\rightarrow \bcV'_j $, where $\phi_{jj'}:=p_{1,V'_{jj'},V_j}$. Then, based on the definition of $\Cov(V_j)$, $\bcV'_j $ is isomorphic to $ g_j^{\ast}\bcV'$ in $\Cov(V_j)$.
\item $\forall j\in J, j'\in J'_j$, we also have the refinement morphisms $\xi_{jj',id}:g_{jj'}^{'\ast}\bcV'_j\rightarrow h_{jj'}^{'\ast}\bcV'$, where
$\xi_{jj'k'}:V'_{jk'}\times_{V_j}V'_{jj'}\rightarrow V'_{jk'}\times_U V'_{jj'}$ is the unique such morphism that makes the following diagram commutative, $\forall j\in J; j',k'\in J'_j$:

$$
\xymatrix{
&&&V'_{jj'}\ar[rd]|{g'_{jj'}}\ar@/^/[drr]^{g'_{jj'}}  &&\\
V'_{jj'}\times_U V'_{jj'}\ar@/^/[urrr]^{p_1}\ar@/_/[drrr]_{p_2}&&V'_{jj'}\times_{V_j} V'_{jj'}\ar[ru]|{p_1}\ar[rd]|{p_2}\ar@{-->}[ll]|{\xi_{jj'k'}}&&V_{j}\ar[r]|{g_{j}}&U\\
&&&V'_{jj'}\ar[ru]|{g'_{jj'}}\ar@/_/[urr]_{g'_{jj'}}&&\\
}
$$
I.e. $\xi_{jj'k'}=id_{V'_{jj'}}\times_{g_j}id_{V'_{jj'}}$. The commutativity of the lower left-hand partial diagram implies that $\xi_{jj',id}$ is a refinement morphism.
\item $\forall j\in J$, consider the commutative diagram, which implies the commutativity of the given diagram:

$$
\xymatrix{\bcP^+(U)\ar[rr]^{\bcP^+(g_j)}&&\bcP^+(V_j)\\
\bcP_U(\bcV')\ar[u]^{j_{\bcV'}}\ar[rr]_{ \bcP_{g_j}(\bcV')}&&\bcP_U(g_j^{\ast}\bcV')\ar[u]^{i_{g_j^{\ast}\bcV'}} \ar[r]_{\tiny \bcP_{V_j}(\phi_{j,id})}&\bcP_{V_j}(\bcV'_j)\ar[ul]_{i_{\bcV'_j}}
}
$$
\end{itemize}
\end{proof}







\subsection{Pre-sheaves on Grothendieck Topology}\label{rho}
\begin{definition}
Let $\bcC, \bcA$ be categories, we define the category of $A$-pre-sheaves on $\bcC$ to be the category $\bcA^{\bcC^{op}}$, and we denote it by $\PreC$.
\end{definition}
Let $\bcP:\bcC^{op}\rightarrow \bcA$ be a pre-sheaf, such that $\bcC$ has Grothendieck pre-topology $\tau$, and $\bcA$ has all products. Then, $\forall\bcU=\{f_i:U_i\rightarrow U, i \in I\}\in Cov_{\tau}(U)$ for $U\in \bcC$:\\
$\forall i\in I,\bcP(f_i):\bcP(U)\rightarrow \bcP(U_i)$, hence $\exists!\rho_{\bcU}:\bcP(U)\rightarrow\displaystyle\prod_{i\in I}\bcP(U_i)$ the unique canonical morphism that makes the following diagrams commute, for all $i\in I$:
$$\xymatrix{
\bcP(U)\ar[dr]^{\bcP(f_i)}\ar@{-->}[d]_{\rho_{\bcU}}&\\
\displaystyle\prod_{k\in I}\bcP(U_k)\ar[r]_{\pi_i}&\bcP(U_i)}$$
\begin{equation}
\pi_i\circ\rho_{\bcU}=\bcP(f_i)
\end{equation} 
$\forall i,j \in I$, let $p_{1,i,j}:=(f_i\times_U f_j)_{U_i},p_{2,i,j}:=(f_i\times_U f_{U_j})_{U_j}$ be the two (in general distinct) pullback projections:
$$\xymatrix{
U_i\times_U U_j\ar[d]_{p_{1,i,j}}\ar[rr]^{p_{2,i,j}}&&U_j\ar[d]^{f_j}\\
U_i\ar[rr]_{f_i}&&U
}$$
Let $\rho_{1,i,j}:=\bcP(p_{1,i,j})$. Notice that, for $i\neq j$ $p_{2,i,j}=p_{1,j,i}$. Hence, $i\neq j$ $\rho_{2,i,j}=\rho_{1,j,i}$.\\
Let $\pi_{i,i}$ be the unique product projection $\pi_{i,i}:\displaystyle\prod_{i,j\in I}\bcP(U_i\times_U U_j)\rightarrow \bcP(U_i\times_U U_i)$, and let $\pi_{i,j},\pi_{j,i}$ be the two (in general distinct) product projections $\pi_{i,j},\pi_{j,i}:\displaystyle\prod_{i,j\in I}\bcP(U_j\times_U U_j)\rightarrow \bcP(U_i\times_U U_j)$, for $i \neq j$.\\
%\tcb{Since, $U_i\times_U U_j=U_j\times_U U_i$, then using the subscript of unordered set is more natural than over-counting the projections by using ordered pairs, or going into the trouble of providing a total ordering on $I$ and distinguishing between $i<j$ and $j<i$.}\\
Then, we define $\rho_{1,\bcU},\rho_{2,\bcU}$ to be the unique morphisms that make the following two diagrams commute:\\

%$${\small\xymatrix{&&\displaystyle\prod_{k\in I}\bcP(U_{k})\ar[dl]_{\pi_i}\ar[dr]^{\pi_j}\ar@{-->}[dd]|{\tcg{\rho_{1,\bcU}}}&&\\ &\bcP(U_i)\ar[dddd]|!{"2,2";"3,2"}{\bcP(p_{1,i,j})}\ar[dddl]_{\tcb{\bcP(p_{1,i,i})}} &&\bcP(U_j) \ar[dddd]|!{"2,4";"3,4"}{\bcP(p_{2,i,j})}\ar[dddr]^{\tcb{\bcP(p_{1,j,j})}}&\\ && \displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l}) \ar[ddll]|{\pi_{i,i}}\ar[dddl]^{\tcr{\pi_{i, j}}} \ar[ddrr]|{\pi_{j,j}}\ar[dddr]_{\tcr{\pi_{j,i}}}&& &&&&\\ &&&&\\ \bcP(U_i\times_U U_i)&&&&\bcP(U_j\times_U U_j)\\ &\bcP(U_i\times_U U_j)&&\bcP(U_i\times_U U_j)&\\}}\$$
$$
\xymatrix{\displaystyle\prod_{k\in I}\bcP(U_{k})\ar@{-->}[dd]_{\tcg{\rho_{1,\bcU}}}\ar[r]^{\pi_i}& \bcP(U_i)\ar[dd]^{\rho_{1,i,j}}\\
\\
\displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l})\ar[r]_{\pi_{i,j}}& \bcP(U_i\times_U U_j)
}
$$
\begin{equation}
\pi_{i,j}\circ \rho_{1,\bcU}=\rho_{1,i,j}\circ \pi_i
\end{equation}

%$${\small\xymatrix{ &&\displaystyle\prod_{k\in I}\bcP(U_{k})\ar[dl]_{\pi_i}\ar[dr]^{\pi_j}\ar@{-->}[dd]|{\tcg{\rho_{2,\bcU}}}&&\\ &\bcP(U_i)\ar[dddd]|!{"2,2";"3,2"}{\bcP(p_{1,i,j})}\ar[dddl]_{\tcb{\bcP(p_{2,i,i})}} &&\bcP(U_j) \ar[dddd]|!{"2,4";"3,4"}{\bcP(p_{2,i,j})}\ar[dddr]^{\tcb{\bcP(p_{2,j,j})}}&\\ && \displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l}) \ar[ddll]|{\pi_{i,i}}\ar[dddl]^{\tcr{\pi_{j,i}}} \ar[ddrr]|{\pi_{j,j}}\ar[dddr]_{\tcr{\pi_{i, j}}}&& &&&&\\ &&&&\\ \bcP(U_i\times_U U_i)&&&&\bcP(U_j\times_U U_j)\\ &\bcP(U_i\times_U U_j)&&\bcP(U_i\times_U U_j)&\\ }} $$
$$
\xymatrix{\displaystyle\prod_{k\in I}\bcP(U_{k})\ar@{-->}[dd]_{\tcg{\rho_{2,\bcU}}}\ar[r]^{\pi_j}& \bcP(U_j)\ar[dd]^{\rho_{2,i,j}}\\
\\
\displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l})\ar[r]_{\pi_{i,j}}& \bcP(U_i\times_U U_j)
}
$$
\begin{equation}
\pi_{i,j}\circ \rho_{2,\bcU}=\rho_{2,i,j}\circ \pi_j
\end{equation}
Hence, we have the diagram $\xymatrix{ \bcP(U)\stackrel{\rho_{\bcU}}{\rightarrow}\displaystyle\prod_{i\in I}\bcP(U_i)\ar@<-3pt>[r]_{\rho_{2,\bcU}}\ar@<3pt>[r]^{\rho_{1,\bcU}} &\displaystyle\prod_{i,j\in I}\bcP(U_i\times_U U_j)}$.\\
When dealing with one covering, the subscript referring to the covering will be omitted as well as it it does not cause confusion.\\
\begin{lemma}\label{p1p=p2p}
Let $\bcP:\bcC_{\tau}^{op}\rightarrow \bcA$ be a pre-sheaf, where $\bcA$ has all products. Let $\rho, \rho_1, \text{ and }\rho_2$ be the above defined canonical morphisms for $\bcU=\{f_i:U_i\rightarrow U|i\in I\}\in Cov_{\tau},U\in \bcC$. Then $\rho_1\circ \rho= \rho_2\circ \rho$.
\end{lemma}
\begin{proof}
Consider the following diagram:
$$
{\small \xymatrix{&&&\bcP(U_i)\ar@/^/[rrrd]^{\rho_{1,i,j}}&&&\\
\bcP(U)\ar@/^/[rrru]^{\bcP(f_i)}\ar@/_/[rrrd]_{\bcP(f_j)}\ar[rr]|{\rho}&&
\displaystyle\prod_{k\in I}\bcP(U_k)\ar@/^/[ru]|{\pi_i}\ar@/_/[rd]|{\pi_j}\ar@<+2pt>[rr]^{\rho_1}\ar@<-2pt>[rr]_{\rho_2}&&\displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l})\ar[rr]|{\pi_{i,j}} &&\bcP(U_i\times_U U_j)\\
&&&\bcP(U_j)\ar@/_/[rrru]_{\rho_{2,i,j}}&&&}}
$$
We notice that the \tcr{external} diagram is commutative, also partial upper and lower diagrams are also commutative.
Hence, $\pi_{i,j}\circ (\rho_1\circ\rho)=\pi_{i,j}\circ (\rho_2\circ\rho)$. Then, the universal property of $\displaystyle\prod_{k,l\in I}\bcP(U_{k}\times_U U_{l})$ implies that $\rho_1\circ\rho=\rho_2\circ\rho$.
\end{proof}
We notice that $\rho_1$ and $\rho_2$ in the above diagram are not necessary different as we see in the below examples:
\begin{example}[$\rho_1=\rho_2$]
Let $\bcC$ be the category $\mathfrak{2}$, i.e. $ \left(\  \xymatrix{I \ar@(ul,ur) \ar[r]^f & T \ar@(ul,ur)}\  \right)$, with initial $I$ and final $T$. $\bcC$ has all pullbacks, hence we consider Grothendieck pre-toplogy $\tau_1(\bcC)$. Then for any functor $\bcP:\bcC^{op}\rightarrow \bcA$ be any functor (an $\bcA$-pre-sheaf on $\bcC$). Then, for  the covering $\{f:I\rightarrow T\}$, we have $\rho=\bcP(f)$, and we have only one pullback diagram:
$$
\xymatrix{I\ar@{-->}[r]^{id_I}\ar@{-->}[d]_{id_I}&I\ar[d]^f\\
I\ar[r]_f&b}
$$
Hence, $\rho_1=\rho_2=id_{\bcP(I)}$.
\end{example}
Below we illustrate minimal non-trivial example with more than one element in the considered covering where $\rho_1=\rho_2$
\begin{example}[$\rho_1=\rho_2$]
Let $\bcC=\bcO(\tau_X)$ be the the preordered category of open subsets of the topological space $(X,\tau_X)$, and let $\bcU=\{f_1:U_1\rightarrow U,f_2:U_2\rightarrow U\}$ be a topological covering of $U$, i.e. $U=U_1\bigcup U_2$. Then, $\rho_1=\rho_2$ for $\bcU$, for any $\bcA$-valued pre-sheaf $\bcP$ such that $\bcP(U_1\bigcap U_2)=T$ the final object of $\bcA$.
\end{example}
\begin{proof}
We have the following pullback diagrams:
$$
\xymatrix{U_1\ar@{-->}[r]^{id_{U_1}}\ar@{-->}[d]_{id_{U_1}}&U_1\ar[d]^{f_1}\\
U_1\ar[r]_{f_1}&U}
\xymatrix{U_1\bigcap U_2 \ar@{-->}[r]^{i_{U_1}}\ar@{-->}[d]_{i_{U_2}}&U_1\ar[d]^{f_1}\\
U_2\ar[r]_{f_2}&U}
\xymatrix{U_2\ar@{-->}[r]^{id_{U_2}}\ar@{-->}[d]_{id_{U_2}}&U_2\ar[d]^{f_2}\\
U_2\ar[r]_{f_2}&U}
$$
Then, in the settings of \ref{rho}, we have:\\
$\bcP(p_{1,1,1})=\bcP(id_{U_1})=\bcP(p_{2,1,1})$\\
$\bcP(p_{1,2,2})=\bcP(id_{U_2})=\bcP(p_{2,2,2})$\\
Since, $\bcP(U_1\bigcap U_2)$ is final in $\bcA$, then $\pi_{1,2}=\pi_{2,1}=t$ the unique morphism $t:\bcP(U_1)\times\bcP(U_1\bigcap U_2)\times\bcP(U_1\bigcap U_2)\times\bcP(U_2)\rightarrow \bcP(U_1\bigcap U_2)$\\
Hence, both diagrams defining $\rho_1$, and $\rho_2$ are identical, and $\rho_1=\rho_2$.
\end{proof}
Notice that when $\rho_1$ differs from $\rho_2$, they differ for two distinct reasons. On the one hand, the existence of two distinct pullback projections in a pullback diagram of the form $U_i\times_U U_i$ in $\bcC$. On the other hand, the existence of two distinct product projections $\bcP(U_i\times_U U_j)\times \bcP(U_i\times_U U_j)\rightarrow \bcP(U_i\times_U U_j)$ in the values category $\bcA$. These are the necessary condition to have $\rho_1\neq \rho_2$.\\
The minimal cases, in terms of number of element of considered covering, of both situation are considered in the below two examples: \tcb{What about minimal in term of number of arrows of $\bcC$!?}
\begin{example}[$\rho_1\neq\rho_2$, due to distinct pullback projections in $\bcC$]
Let $\bcC$ be the category with all pullbacks (e.x. $\Sets$) that has the below pullback graph:
$$\xymatrix{
W\ar[r]^g\ar[d]_h&V\ar[d]^f\\
V\ar[r]_f&U
}$$
such that $g\neq h$ (e.x. for $\bcC=\Sets$, $U$ is final object, then $W=V\times V$, $g,h$ are the two distinct product projection, for $|V|>1$.)\\
Since $\bcC$ has all pullbacks, then we may consider Grothendieck pre-topology $\tau_1(\bcC)$, and let $\bcP:\bcC_{\tau_1(\bcC)}^{op}\rightarrow \bcA$ be a pre-sheaf on $\bcC$.\\
Then, for the covering $\{f:V\rightarrow U\}\in Cov_{\tau_1(\bcC)}(U)$, let $f_1=f,U_1=V$, then, in the settings of \ref{rho}, we have: $\rho=\bcP(f)$, and $p_{1,1,1}=g, p_{2,1,1}=h$, (or the other way round), hence\\
$\rho_1  =id_{\bcP(W)}\circ\rho_1=\bcP(g)\circ id_{\bcP(V)}=\bcP(g)$\\
$\rho_2  =id_{\bcP(W)}\circ\rho_2=\bcP(h)\circ id_{\bcP(V)}=\bcP(h)$\\
Then for the pre-sheaves that satisfy $\bcP(g)\neq\bcP(h)$, we have $\rho_1\neq \rho_2$. \tcb{We notice that $\bcP$ is a sheaf}
\end{example}
\begin{example}[$\rho_1\neq\rho_2$, due to distinct product projections in $\bcA$]
Let $\bcC=\bcO(\tau_X)$ be the preordered category of open subsets of the topological space $(X,\tau_X)$. Let $\bcU=\{f_1:U_1\rightarrow U, f_2:U_2\rightarrow U\}\in Cov_{\tau_X}(U), U\in \bcC$, a topological covering for $U$, i.e $U=U_1\cup U_2$, where $f_1$ and $f_2$ are the unique such inclusion morphisms.\\
Let $\bcP:\bcC_{\tau_X}^{op}\rightarrow \bcA$ be an $\bcA$-pre-sheaf on $\bcC$, for some concrete category $\bcA$, such that $\bcP(U_1\bigcap U_2)$ is not the final object of $\bcA$.\\
Then, $\rho$ is the unique morphism that makes the following diagram commutative:\\
$${\small \xymatrix{\bcP(U)\ar[dd]_{\bcP(f_1)}\ar[rr]^{\bcP(f_2)}\ar@{-->}[ddrr]|{\tcg{\rho}}&&\bcP(U_2)\\
\\
\bcP(U_1)&&\bcP(U_1)\times \bcP(U_2)\ar[ll]^{\pi_1}\ar[uu]_{\pi_2}
}}$$
Where $\pi_1((x_1,x_2))=x_1$, and $\pi_2((x_1,x_2))=x_2, \forall (x_1,x_2)\in \bcP(U_1)\times \bcP(U_2)$ (we could have defined them the other way round). Hence, $\rho(x)=(\bcP(f_1)(x),\bcP(f_2)(x)),\forall x \in \bcP(U)$.\\
In order to induce $\rho_1$ and $\rho_2$, we consider the below pullback diagrams:
\begin{center}
$\xymatrix{U_1\ar@{-->}[r]^{id_{U_1}}\ar@{-->}[d]_{id_{U_1}}&U_1\ar[d]^{f_1}\\
U_1\ar[r]_{f_1}&U}$
$\xymatrix{U_1\bigcap U_2\ar@{-->}[r]^{i_1}\ar@{-->}[d]_{i_2}&U_1\ar[d]^{f_1}\\
U_2\ar[r]_{f_2}&U}$
$\xymatrix{U_2\ar@{-->}[r]^{id_{U_2}}\ar@{-->}[d]_{id_{U_2}}&U_2\ar[d]^{f_2}\\
U_2\ar[r]_{f_2}&U}$
\end{center}
Where $i_1,i_2$ are the unique such inclusion morphisms.\\
Then, $\rho_1$ is the unique morphism that makes the following diagram commutative:\\
$$\xymatrix{
&&\bcP(U_1)\times \bcP(U_2)\ar[ld]_{\pi_1}\ar[rd]^{\pi_2}\ar@{-->}[dd]|{\rho_1}&&\\
&\bcP(U_1)\ar[ldd]_{id_{\bcP(U_1)}}\ar[dd]^{\bcP(i_1)}&& \bcP(U_2)\ar[rdd]^{id_{\bcP(U_2)}}\ar[dd]_{\bcP(i_2)}&\\
&&  \displaystyle\prod_{i,j\in \{1,2\}}\bcP(U_i\bigcap U_j) \ar[lld]_{\pi_1}\ar[ld]^{\tcr{\pi_2}}\ar[rd]_{\tcr{\pi_3}}\ar[rrd]^{\pi_4}&&\\
\bcP(U_1)&\bcP(U_1\bigcap U_2)&&\bcP(U_1\bigcap U_2)& \bcP(U_2)
}$$
Where $\pi_i((x_1,x_2,x_3,x_4))=x_i,i=1..4$. Therefore, $\rho_1((x_1,x_2))=(x_1,\bcP(i_1)(x_1),\bcP(i_2)(x_2) ,x_2)$.\\
On the other hand, $\rho_2((x_1,x_2))=(x_1,\bcP(i_2)(x_2) ,\bcP(i_1)(x_1),x_2)$. That, $\rho_2$ is the unique morphism that makes the following diagram commutative:\\
$$\xymatrix{
&&\bcP(U_1)\times \bcP(U_2)\ar[ld]_{\pi_1}\ar[rd]^{\pi_2}\ar@{-->}[dd]|{\rho_1}&&\\
&\bcP(U_1)\ar[ldd]_{id_{\bcP(U_1)}}\ar[dd]^{\bcP(i_1)}&& \bcP(U_2)\ar[rdd]^{id_{\bcP(U_2)}}\ar[dd]_{\bcP(i_2)}&\\
&&  \displaystyle\prod_{i,j\in \{1,2\}}\bcP(U_i\bigcap U_j) \ar[lld]_{\pi_1}\ar[ld]^{\tcr{\pi_3}}\ar[rd]_{\tcr{\pi_2}}\ar[rrd]^{\pi_4}&&\\
\bcP(U_1)&\bcP(U_1\bigcap U_2)&&\bcP(U_1\bigcap U_2)& \bcP(U_2)
}$$
Then, $\rho_1\neq \rho_2$ as well as either of $\bcP(i_2)$ and $\bcP(i_1)$ is not a constant morphism, or if they are different constant morphism in $\bcA$. However, straightforward calculations shoes that $\rho_1\circ \rho= \rho_2\circ \rho$.
\end{example}
The same argument stands for any preordered category, for any covering with at least two elements. However, if $\bcC$ is not preordered, we still might get $\rho_1\neq\rho_2$, for some coverings, but it wont necessary be based on the sole reason of having distinct product projection in $\bcA$.\\
The above two conditions (in the last two examples) are necessary to have $\rho_1\neq \rho_2$. That, if neither of them is satisfied, then $\rho_1$ and $\rho_2$ would be generated from the same diagram, and they would be equal! \tcb{However, one should notice that these conditions are not sufficient to guarantee the difference, as we see below: type the example}.\\
\subsection{Sheaves}
\begin{definition}[Sheaf]
Let $\bcS:\bcC_{\tau}^{op}\rightarrow \bcA$ be a pre-sheaf, where $\bcA$ has all products. We say that $\bcS$ is a sheaf iff $\forall\{f_i:U_i\rightarrow U, i \in I\}\in Cov_{\tau}(U)$ for $U\in \bcC$, the following diagram is exact:\\
\begin{equation}\label{Sheaf}
\xymatrix{ \bcS(U)\stackrel{\rho}{\rightarrow}\displaystyle\prod_{i\in I}\bcS(U_i)\ar@<-3pt>[r]_{\rho_2}\ar@<3pt>[r]^{\rho_1} &\displaystyle\prod_{i,j\in I}\bcS(U_i\times_U U_j)}
\end{equation}
Where $\rho, \rho_1, \text{ and }\rho_2$ are the above defined canonical morphisms in \ref{rho} .
\end{definition}
\begin{definition}
Let $\bcC_{\tau}$ be a site, $\bcA$ be a category with all products, we define the category of $A$-sheaves on $\bcC_{\tau}$ to be the full subcategory of sheaves in $\PreC$, and we denote it by $\ShvC$.
\end{definition}
\begin{remark}
Let $\bcS:\bcC_{\tau}^{op}\rightarrow \bcA$ be a sheaf, since \ref{Sheaf} is exact, then $\rho$ is a monomorphism. We will distinguish a subcategory of pre-sheaves that satisfy this condition, as in the following definition. This subcategory will play an important rule later on in the procedure of sheafification \ref{Sheafification}.
\end{remark}
\begin{definition}[Separated Pre-sheaf]
Let $\bcP:\bcC_{\tau}^{op}\rightarrow \bcA$ be a pre-sheaf, where $\bcA$ has all products. We say that $\bcP$ is a separated pre-sheaf iff $\forall\bcU=\{f_i:U_i\rightarrow U, i \in I\}\in Cov_{\tau}(U)$ for $U\in \bcC$, then the canonical morphisms $\rho: \bcP(U) \rightarrow \displaystyle\prod_{i\in I}\bcP(U_i) $, defined in \ref{rho}, is a monomorphism.
\end{definition}
\begin{fact}
Concrete valued sheaves over the above mentioned Grothendieck pre-topology $\tau$ on the pre-order category $\bcC$, of open subsets of a topological space $(U,\tau_U)$, coincide with sheaves over the the topological space $(U,\tau_U)$.
\end{fact}
\begin{proof}
\tcg{Write the proof}.
\end{proof}
\begin{example}
\tcb{Write an example of a pre-sheaf that is not a sheaf distinguishing between different causes}.\\
\tcb{Write examples of a sheaves}.
\end{example}
\tcr{What's the advantage of sieves and topologies over pre-topology!?}\\
\begin{example}(Canonical Topology on a category, a natural example of Sheaves)
Let $\bcC$ be a category with all pullbacks, we notice that the definition of family of effective epimorphisms \ref{Effective epimorphism} that they are the covering-like family of morphisms that for which all representable pre-sheaves satisfy the axiom of sheaf. Therefore, one might be interested in forming a topology of these families, so that all representable pre-sheaves becomes sheaves with respect to this topology! We notice that in general the collections of these these families does not satisfy the axioms of Grothendieck pre-topology. \tcb{Give an example! It should violate condition 2, just give an example for FEE which is not FUEE!}. However, this issue is overcome be considering the families of universal effective epimorphisms \ref{Universal Effective epimorphism} rather than the families of effective epimorphism. We will refer to a family of [universal] effective epimorphisms by F[U]EE for short\label{FUEE}.\\
Let $\tau_{Can}=\{Cov_{Can}(U)|U\in \bcC\}$, where $Cov_{Can}(U)$ is the collection of families of universal effective epimorphisms of the form $\bcU=\{f_i:U_i\rightarrow U|i\in I\}$. Then, $\tau$ defines a Grothendieck pre-topology on $\bcC$. \tcb{in fact it is a topology}. Furthermore, all representable pre-sheaves on $\bcC_{\tau}$ are sheaves.
\end{example}
\begin{proof}
$\tau$ is a pre-topology on $\bcC$, that:
\begin{itemize}
\item Identity axiom: $\forall U\in \bcC$, let $\bcU=\{id_U\}$, then for any $f:V\rightarrow U$ in $\bcC$ we have the following pullback diagram:
$$
\xymatrix{V\ar@{-->}[r]^{id_V}\ar@{-->}[d]_{f}&V\ar[d]^f\\
U\ar[r]_{id_U}&U}
$$
Then $\bcU_f=\{id_V\}$ which is a FEE. Hence, $\{id_U\}$ is a FUEE.
\item Stability axiom: $\forall f:V\rightarrow U, \bcU=\{f_i:U_i\rightarrow U:i \in I\}\in Cov_{Can}(U)$ , then the family $\bcV:=\bcU_f=\{\pi_V:U_i\times_{U}V\rightarrow V,i \in I\}$ exists, that $\bcC$ has all pullbacks, then for any $g:W\rightarrow V$ in $\bcC$ we have the follwoing pullback diagrams:
$$
\xymatrix{U}
$$
\tcg{finish the proof}
 and is a covering of $V$, where $(f_i\times_U f)_{V}$ is the projection, on $V$, of the pullback of the diagram $U_i\rightarrow U \leftarrow V$, for each $i \in I$.
\item Transitivity axiom: \tcg{finish the proof} $\forall \{f_i:U_i\rightarrow U,i \in I\}\in Cov_{\tau}(U), 
\{g_{i,j}:U_{i,j}\rightarrow U_i:j\in J_i\}\in Cov_{\tau}(U_i)$ for $i \in I$, then $\{f_i\circ g_{i,j}:j\in J_i, i\in I\}\in Cov_{\tau}(U)$.
\end{itemize}
The rest of the proof is a automatic based on the definition of sheaves and FUEE.
\end{proof}
$\tau_{Con}$ is the finest topology on which all representable pre-sheaves are sheaves. That, let $\tau'$ be a topology on $\bcC$ for which all representable pre-sheaves are sheaves, then every covering of covering family of $\tau'$ is a FUEE, i.e. a covering family of $\tau_{Con}$.

%!!\subsection{Sheafification}
%\tcg{??!!}
%\tcb{Is sheafification always possible? No, we need to substitute the kernel, by equaliser.}
\subsection{Continuous maps of sites}
\noindent Let $\bcC_{\tau},\bcD_{\tau'}$ be sites, $f:\bcC\rightarrow\bcD$ be a functor of the underlying categories, we say that $f$ is a map of sites if it satisfies the following conditions:
\begin{itemize}
\item 
\item The canonical morphism $f(U_i)\times_{f(X)}f(U_j)\rightarrow f(U_i\times_{X}U_j)$ is an isomorphism.
\end{itemize}


\black
